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"... For any quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds ..."

For any quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar−1. The main results are that we resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

"... In [23, 24], Y.-P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.-P. Lee conjectured that the two sets of relations coincide and proved ..."

In [23, 24], Y.-P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.-P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.-P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.-P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it implies that in any semi-simple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov–Witten potentials coincide.

...isfied certain postulated properties. The space of r-spin structures and the class cW were later constructed precisely and shown to possess the expected properties by a joint effort of several people =-=[15, 16, 27, 31, 30]-=-. 2. A. Givental [8] constructed a transitive group action on all semi-simple formal Gromov–Witten theories. He found a specific group element that takes the Gromov–Witten potential of a point to the ...

"... Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main ..."

Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action. 1.

...In particular, Jarvis–Kimura–Vaintrob [13] established the genus zero case of the conjecture; T. Mochizuki and A. Polishchuk independently established the following property for τ r-spin : Theorem 1. =-=[20, 23]-=- All tautological equations hold for F r-spin g . 1 satisfies all “expected functorial properties”, similar to the axioms formulated by Kontsevich–Manin in the Gromov–Witten theory. However, Riemann’s...

"... Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward constructi ..."

Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward construction via K-theory. In this way we short-circuit the passage through bivariant intersection theory and the use of MacPherson’s graph construction. Furthermore, we show that the Witten top Chern class admits a natural lifting to the K-theory ring. 1.

...e Chow ring, they obtain a Chow cohomology class cPV (see Section 5.2). Let χ denote χ(C,L ); we have cPV ∈ A −χ (X)Q. Furthermore, such a class is compatible with Witten’s earlier definition and, by =-=[Pol04]-=-, satisfies all the axioms of the cohomological field theory defined in [JKV01]—this is a preliminary condition to Witten’s conjecture [Wit93]. 1.5. Our construction of the Witten top Chern class. We ...

...hierarchies of systems of partial differential equations (see [9]). The first example of CohFTs besides GW-theory was provided by the theory of r-spin curves constructed in [60], [61], [23], [48] and =-=[47]-=- (see also [42]). The corresponding Frobenius manifolds are isomorphic to the ones constructed by Saito for simple singularities of type Ar−1, and the corresponding integrable hierarchies are the Gelf...

"... Abstract. We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasihomogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the ..."

Abstract. We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasihomogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W-spin equations when W = W(x1,..., xt) is a nondegenerate quasi-homogeneous polynomial with fractional degrees (or weights) wt(xi) = qi &lt; 1/2 for all i. In particular, the compactness theorem holds for the superpotentials E6, E7, E8, and An−1, Dn+1 for n ≥ 3. 1.

...ibution of a Ramond marked point to the corresponding field theory is zero in the Ar−1 case (the decoupling of the Ramond sector). This was proved true for genus zero in [JKV] and for higher genus in =-=[P]-=-. This is partly why the moduli space of spin curves has been around for a long time while the spin equation seems to have been lost in the literature. In the course of our investigation, we discovere...

"... Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten’s class over th ..."

Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten’s class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals. This allows us, using the methods of [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the ψ-classes (or tautological classes) over any moduli space of r-spin structures, in short, all numbers involved in Witten’s conjecture. 1

"... The exact FZZT brane partition function for topological gravity with matter is computed using the dual two-matrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topolo ..."

The exact FZZT brane partition function for topological gravity with matter is computed using the dual two-matrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topological gravity. Using the well-known relation between the Kontsevich integral and a certain shift in the closed-string background, we conclude that these models exhibit open/closed string duality explicitly. Just as in pure topological gravity, the unphysical sheets of the classical FZZT moduli space are eliminated in the exact answer. Instead, they contribute small, nonperturbative corrections to the exact answer through Stokes ’ phenomenon. January

"... Using Picard–Lefschetz periods for the singularity of type AN, we construct a projective representation of the Lie algebra of differential operators on the circle with central charge h: = N + 1. We prove that the total descendant potential DAN of AN-singularity is a highest weight vector. It is kno ..."

Using Picard–Lefschetz periods for the singularity of type AN, we construct a projective representation of the Lie algebra of differential operators on the circle with central charge h: = N + 1. We prove that the total descendant potential DAN of AN-singularity is a highest weight vector. It is known that DAN can be interpreted as a generating function of a certain class of intersection numbers on the moduli space of h-spin curves. In this settings our constraints provide a complete set of recursion relations between the intersection numbers. Our methods are based entirely on the symplectic loop space formalism of A. Givental and therefore they can be applied to the mirror models of symplectic manifolds.